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fundamental development cannot be over-estimated. Gauss,
Abel, and Jacobi, by their extensive and detailed elaboration
of the theory of elliptic functions, in which complex numbers
appear inevitably, provided a testing ground for the discovery
and improvement of general theorems in the theory of functions
of a complex variable. The two theories seemed to have been
designed by fate to complement and supplement one another -
there is a reason for this, also for the deep connexion of elliptic
functions with the Gaussian theory of quadratic forms, which
considerations of space force us to forego. Without the innu-
merable clues for a general theory provided by the special
instances of more inclusive theorems occurring in elliptic f unc-
tionsj the theory of functions of a complex variable would have
developed much more slowly than it did - Liouville's theorem,
the entire subject of multiple periodicity with its impact on
the theory of algebraic functions and their integrals, may be
recalled to mathematical readers. If some of these great monu-
ments of nineteenth-century mathematics are already receding
into the mists of yesterday, we need only remind ourselves that
Picard's theorem on exceptional values in the neighbourhood
of an essential singularity, one of the most suggestive in current
analysis, was first proved by devices originating in the theory
of elliptic functions. With this partial summary of the reason
why elliptic functions were Important in the mathematics of
the nineteenth century we may pass on to Jaeobi's cardinal
part in the development of the theory.
The history of elliptic functions is quite involved, and
although of considerable interest to specialists, is not likely to
appeal to the general reader. Accordingly we shall omit the
evidence (letters of Gauss, Abel, Jacobi, Legendre, and others)
on which the following bare summary is based-
First, it is established that Gauss anticipated both Abel and
Jacobi by as much as twenty-seven years in some of their most
stoking work. Indeed Gauss says that 'Abel has followed
exactly the same road that I did in 1798'. That this claim is just
wiH be admitted by anyone who will study the evidence pub-
lished only after Gaups' death. Second, it seems to be agreed
that Abel anticipated Jacobi in certain important details, but